Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi)}) ^ {8} $
Solution: Let's express our complex number in Euler form first. $ {\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi)} = { e^{\pi i / 3}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{\pi i / 3}}) ^ {8} = e ^ {8 \cdot (\pi i / 3)} $ The angle of the result is $8 \cdot \frac{1}{3}\pi$ , which is $\frac{8}{3}\pi$ Our result is $ e^{2\pi i / 3}$. Converting this back from Euler form, we get $\cos(\frac{2}{3}\pi) + i \sin(\frac{2}{3}\pi)$.